Triaxial leak criterion with quadratic dependence on effective pressure for optimizing threaded connections in well tubulars

ABSTRACT

Example systems, devices, and methods are described for evaluating the relative leak safety of threaded connections in well tubulars. The method includes evaluating leak risk using a quadratic leak criterion, which is a function of the effective pressure (Pe) and includes three constants: a first constant called the leak path factor (δ), a thread modulus (α), and a makeup leak resistance (β). The method in some implementations includes identifying at least three test cases with different Δp expected to leak under certain conditions. The method includes fitting a quadratic expression to a subset of values associated with the leaking test cases, in which the resulting quadratic expression includes values for the three constants. The quadratic leak criterion can be expressed as a leak safety factor for evaluation of particular threaded connections, load cases, and conditions, thereby optimizing the design and selection of threaded connections and also optimizing connection performance during drilling and production operations.

CROSS REFERENCE TO RELATED APPLICATIONS

The present application claims the benefit of and priority to U.S. Provisional Application 63/263,032 filed Oct. 26, 2021, entitled, “Triaxial Leak Criterion for Connections Based on Leak Limit with Hydrostatic Dependence.” The present application is a continuation-in-part of currently pending U.S. application Ser. No. 17/452,064 filed Oct. 23, 2021, which is a continuation-in-part of U.S. application Ser. No. 16/413,331 filed May 15, 2019 (now U.S. Pat. No. 11,156,256), which claims the benefit of and priority to U.S. Provisional Application 62,671,771 filed May 15, 2018. The entirety of each identified application is incorporated herein by reference.

TECHNICAL FIELD

Examples set forth in the present disclosure relate to the design of well tubulars and the optimization of threaded connections in well tubulars. More particularly, but not by way of limitation, the present disclosure describes using a quadratic leak criterion with three constants and an associated leak safety factor to evaluate the relative safety of threaded connections.

BACKGROUND

During the drilling process, a well hole segment is typically supported by a tubular pipe called a casing. The next casing inside the previous casing supports the next hole segment, and so on, until the casing reaches the total depth of the well. The innermost pipe is the production string called tubing. The term ‘tubular’ will be used here for all wellbore pipes, including tubing, casing, liners, and tiebacks, as well as drill pipe, bottom hole assemblies, and work strings. In most strings, all the joints and connections are the same; namely, the same pipe diameters (inner and outer), the same connection type and size, and the same yield strength. Tapered strings have two or more sections in which all the joints and connections are the same.

Most well tubulars are fabricated in cylindrical sections (typically, thirty to forty feet in length), oriented in a column, and joined together with threaded connections. The size, wall thickness, and yield strength of the tubular and its threaded connections are structurally designed to reduce the risk of failure, including pipe-body failure (burst, collapse, or axial parting) and threaded connection failure (leak, pull-out, or thread fracture).

For pipe body design, the yield strength (Y) of the pipe material (steel, typically) can be measured by placing a rod-shaped specimen of the material under axial tension until it yields. To evaluate pipe integrity, the yield point of the pipe body is typically compared to the von Mises stress (σ_(VM)), which is a scalar quantity determined from the stresses acting on the pipe body. In general, if the von Mises stress (σ_(VM)) is less than the yield strength (Y), the pipe body is safe.

For connection design, connection failure due to leak is an important part of current tubular design in the petroleum industry. The threaded connections currently available include standard connections (sometimes called API connections, referring to the standards of the American Petroleum Institute) and proprietary designs sometimes called premium connections. The mechanical behavior of a threaded connection can be estimated by using finite element analysis (FEA) or by conducting extensive testing. Current qualification testing procedures for leak assessment of a threaded connection involve numerous laboratory tests, conducted using specified axial forces (tension or compression) and specified internal and external pressures. The test procedure is repeated for a variety of different conditions, using multiple specimens of the connection. Each specimen is tested until it leaks, or until the pipe (or the connection) fails. Internal leak tests are performed with no external backup pressure. External leak tests are performed with no internal backup pressure. The total cost of such testing can exceed one million dollars—and the results apply only to the specific threaded connection tested, and only to the precise size tested. Additional FEA or laboratory testing must be conducted for other sizes of the same connection design. The test results are typically plotted on a graph to display a connection usage envelope; analogous to the von Mises ellipse, with pressure on the ordinate (internal pressure positive) and axial force on the abscissa (tension positive). If the estimated loads on a threaded connection fall within the leak envelope, the connection is safe.

For tubular design, including the design for threaded connections, the currently used design criteria do not include a consideration of hydrostatic pressure. The current failure theories and equations for both the pipe body and for connections are based on shear and ignore hydrostatic pressure.

BRIEF DESCRIPTION OF THE DRAWINGS

Features of the various implementations disclosed will be readily understood from the following detailed description, in which reference is made to the appending drawing figures. A reference numeral is used with each element in the description and throughout the several views of the drawing. When a plurality of similar elements is present, a single reference numeral may be assigned to like elements, with an added lower-case letter referring to a specific element.

The various elements shown in the figures are not drawn to-scale unless otherwise indicated. The dimensions of the various elements may be enlarged or reduced in the interest of clarity. The several figures depict one or more implementations and are presented by way of example only and should not be construed as limiting. Included in the drawings are the following figures:

FIG. 1 is a graphical representation of a triaxial connection leak criterion, plotted as a leak line;

FIG. 2 is a graphical representation of a triaxial connection leak criterion, plotted as a leak circle;

FIG. 3 is a graphical representation of a leak safety factor for a particular string, including a load case line for a number of load cases;

FIG. 4 is a cross-sectional view of an example tubing or casing coupled connection having a radial interference seal;

FIG. 5 is a cross-sectional view of an example drill string connection having an axial interference shoulder seal;

FIG. 6 is a graphical representation of the predicted leak conditions for an example API LTC connection;

FIG. 7 is a graphical representation of the predicted failure conditions for a pipe-body alone and for the example API LTC connection;

FIG. 8 is a partial cross-sectional view of an example generic premium connection;

FIG. 9 is a table of values associated with a plurality of example leaking test cases;

FIG. 10 is a graph of the leaking test cases from FIG. 9 and a quadratic expression fitted to the values; and

FIG. 11 is a graphical representation of a quadratic leak safety factor for an example string, including example load case lines for a number of example load cases.

DETAILED DESCRIPTION

Various implementations and details are described with reference to an example method of calculating a leak safety factor associated with a threaded connection as a function of effective pressure (Pe=P−σn) and von Mises stress (σ_(VM)). The leak safety factor includes a quadratic expression with three constants: a leak path factor (δ), a thread modulus (α), and a makeup leak resistance (β). The method includes approving the threaded connection if the leak safety factor is greater than a threshold value. In some implementations, the method includes generating at least three leaking test cases with different Δp=Pi−Po, each characterized by a combination of loads expected to result in leak at the threaded connection, to determine the three leak constants. For each leaking test case, the method includes calculating a subset of values including an effective pressure value (Pe) and a von Mises stress value (σ_(VM)). The method includes fitting a quadratic expression to the subset of values. The resulting quadratic expression includes values for the three constants: the leak path factor (δ), the thread modulus (α), and the makeup leak resistance (β).

The following detailed description includes numerous details and examples that are intended to provide a thorough understanding of the subject matter and its relevant teachings. Those skilled in the relevant art may understand how to apply the relevant teachings without such details. This disclosure is not limited to the specific devices, systems, and methods described because the relevant teachings can be applied or practiced in a variety of ways. The terminology and nomenclature used herein is for the purpose of describing particular aspects only and is not intended to be limiting.

Those skilled in the relevant art will recognize and appreciate that many changes can be made to the various aspects of the implementations described herein, while still obtaining the beneficial results. It will also be apparent that some of the desired benefits can be obtained by selecting some of the features, but not others. Accordingly, those who work in the art will recognize that many modifications and adaptations are possible and may even be desirable in certain applications, and that these are part of the disclosure.

The terms “comprising” and “including,” and any forms thereof, are intended to indicate a non-exclusive inclusion; that is, to encompass a list that includes the items listed and may include others not expressly listed. As used herein, the singular forms “a,” “an” and “the” include plural referents unless the context clearly dictates otherwise. Thus, for example, reference to a component can include two or more such components unless the context indicates otherwise. Relational terms such as “first” and “second” and the like may be used solely to distinguish one element or action from another, without implying any particular order between such elements or actions.

The terms “optional” or “optionally” mean that the subsequently described element or action may or may not occur. In other words, such a description includes instances where the element or action occurs and instances where it does not.

Then term “facilitate” means to aid, assist, enable, improve, or make easier. The term “inhibit” means to hinder, restrain, impede, restrain, thwart, oppose, or obstruct.

The words “proximal” and “distal” are used to describe items or portions of items that are situated closer to and away from, respectively, a user or a viewer. Thus, for example, the near end or other portion of an item may be referred to as the proximal end, whereas the generally opposing portion or far end may be referred to as the distal end.

As used herein, the term “constant” refers to a quantity or value that represents an inherent property of an object or thing. As opposed to a variable, a constant associated with an object does not change in response to extrinsic states or conditions. For example, the yield strength (Y) of a particular material is a constant.

Currently, the role of hydrostatic pressure in the design of well tubulars is not fully appreciated in the current failure theories for yield, buckling, connection leak, pull-out, or thread fracture. Hydrostatic pressure refers to the pressure exerted by a fluid or fluid-like substance on a body immersed in the substance. The currently used design criteria, failure theories, and equations for tubular design do not depend on hydrostatic pressure.

The total stress at each point in a tubular can be divided into a shear part and a hydrostatic part. The von Mises stress, defined as a scalar measure of the shear part, is used to assess the yield of the pipe material or, more specifically, to assess burst, collapse, tensile failure, and compressive failure of the pipe-body. The Lubinski fictitious force, often called the effective force, is used to evaluate buckling. Both yield and buckling are independent of the hydrostatic part of the stress; i.e., shear behavior alone governs these design criteria.

As part of tubular design, threaded connections, especially proprietary connections, are often evaluated and qualified for strength and sealing with design plots, called connection usage or service envelopes. These envelopes are used similarly to the von Mises ellipse and have the same axes, namely axial force and differential pressure. Laboratory tests and/or FEA (finite element analysis) are performed to develop the envelopes. The leak tests are typically performed with no backup pressure and with loads up to some percent of von Mises yield of the pipe body, so the effect of hydrostatic pressure is not even considered, implying that connection leak depends only on the shear part of the stress, like the pipe body.

The leak envelope for a threaded connection based on laboratory test results, if available, is used for tubular design by assuming that the internal or external pressure can be replaced with the differential pressure. This assumption is not supported by experimental evidence, creates inaccuracy, and eliminates the effect of hydrostatic pressure. Because the hydrostatic effect is disregarded, current connection design is biaxial (at best).

NEW APPROACH: Contrary to current practice, and as demonstrated herein, the leak resistance of a threaded connection is directly dependent on hydrostatic pressure. Hence, leak resistance is a function of the threaded connection's location (depth) in the well string. Tubular analysis using the von Mises equations alone (which assume shear only) does not adequately characterize the risk of leak at a threaded connection.

A new triaxial connection leak criterion is described herein, which includes the effect of hydrostatic pressure in a linear equation. Based on the new leak criterion, a new leak connection safety factor is described for evaluating connections and designing tubulars. A leak line and a leak circle are also described, which provide a graphical representation to quickly identify whether the expected loads are critical and will cause leak.

The tubular stresses can be calculated at every depth (z) based on the internal pressure (Pi), external pressure (Po), and axial force (Fz). (For clarity in small print, these and other denoted variables may appear herein without subscript letters.) This set of conditions—internal pressure (Pi), external pressure (Po), and axial force (Fz)—is known or can be measured or determined, in most cases, including any effect from temperature. As used herein, the depth (z) represents the axial or longitudinal direction along a tubular string, whether the well is solely vertical or not. For a vertical well, the measured depth equals the true vertical depth. For a non-vertical well, the measured depth is greater than the true vertical depth.

The complete description of the stress acting at a material point on a body is a nine-component tensor, which can be expressed as a three-by-three matrix (often depicted as a cube, with one normal stress and two shear stresses on each of the six faces; the stresses on opposing faces are equal). For a well tubular, in cylindrical coordinates, the normal stresses include the radial stress (σr), the hoop stress (σθ), and the axial stress (σz). The radial stress and hoop stress depend only on the internal pressure (Pi), external pressure (Po), and the inner and outer diameter of the pipe, and can be calculated using the Lame equations. The axial stress can be calculated from the axial force (Fz), which includes pipe weight, and the cross-sectional area of the pipe. Most tubular design does not consider shear stresses, such as the stress generated by friction between the outer pipe surface and the borehole wall.

HYDROSTATIC PRESSURE: The leak resistance of a threaded connection is directly dependent on hydrostatic pressure. The mean normal stress, P, which is defined as the scalar measure of the hydrostatic part of the stress, is equal to the average of the three normal stresses, as follows, in cylindrical coordinates for a tubular:

$P = {\frac{\sigma_{r} + \sigma_{\theta} + \sigma_{z}}{3}.}$

In fact, by definition, the sum of the three normal stresses is simply the first stress invariant, which itself is a scalar and independent of the coordinate system. This coordinate independence (just as with the von Mises stress, which is the second invariant of the shear part of the stress) is an important feature of the new triaxial leak criterion. In solid mechanics and as used herein, tensile stress is considered positive and compressive stress is negative. The term ‘hydrostatic pressure’ as used herein is equivalent to negative of the mean normal stress, P, which reduces to the conventional meaning of hydrostatic pressure in certain circumstances; e.g., at the bottom of an open-end tubular in a drilled hole filled with static fluid, all three normal stresses in the tubular at its base are compressive and equal to the negative of the fluid pressure at that depth. In most cases, the three normal stresses at any point along the string and within the tubular wall are not equal. Hence, the mean normal stress, P, is triaxial and generalizes the hydrostatic concept to three dimensions.

TRIAXIAL CONNECTION LEAK CRITERION: The leak behavior of a threaded connection under triaxial stress is fully described by the new connection leak criterion 25: σ_(VM) =−αP+β  “Triaxial Leak Criterion 25” where P is the mean normal stress, and where alpha (α) and beta (β) are new connection constants, as described and defined herein. The mean normal stress, P, in the Triaxial Leak Criterion 25 introduces the dependence of leak on hydrostatic pressure—which is fundamental and different from all prior leak theories.

The new Triaxial Leak Criterion 25 is also the basis for a new triaxial leak connection safety factor:

${SF_{Leak}} = \frac{{{- \alpha}P} + \beta}{\sigma_{VM}}$ in which the numerator represents the working limit and the denominator represents the working stress. In practice, as described herein, a designer can approve a threaded connection if the leak safety factor is greater than a threshold value, which is typically a number equal to or greater than one. An additional ‘design factor’ may be selected by the designer (or imposed by an operator) as an additional margin of safety, in order to account for manufacturing tolerances and the like. For example, a designer or operator may specify 1.25 for the threshold value (which is the number typically used for the pipe body).

The connection constants, alpha (α) and beta (β), represent inherent properties of a particular threaded connection, independent of extrinsic states or conditions. The connection constants for a particular connection are analogous to the material constant for a particular material; for example, the yield strength constant (Y) for a pipe-body material (steel, typically). In fact; if alpha (α) equals zero, the hydrostatic dependence is eliminated and the Triaxial Leak Criterion reduces identically to the von Mises criterion (similar to that for the pipe body), and the constant beta (β) is analogous to the yield strength (Y).

Using the known equations (without shear) for von Mises stress (σ_(VM)) and for mean normal stress (P), the Triaxial Leak Criterion becomes:

$\sigma_{VM} = {\sqrt{\frac{1}{2}\left\lbrack {\left( {\sigma_{r} - \sigma_{\theta}} \right)^{2} + \left( {\sigma_{\theta} - \sigma_{z}} \right)^{2} + \left( {\sigma_{z} - \sigma_{r}} \right)^{2}} \right\rbrack} = {{{- \alpha}\frac{\left( {\sigma_{r} + \sigma_{\theta} + \sigma_{z}} \right)}{3}} + {\beta.}}}$

Using the expressions for the axes of the von Mises circle and the Lubinski formula for neutral stress (σ_(n)), the Triaxial Leak Criterion becomes:

$\sqrt{(\Delta)^{2} + \left( {\Delta\sigma}_{z} \right)^{2}} = {{{- \alpha}\frac{{2\sigma_{n}} + \sigma_{z}}{3}} + \beta}$ where ΔP is the y-axis and Δσ_(z) is the x-axis of the von Mises circle, and where the mean normal stress (P) is expressed as a function of the Lubinski neutral stress (σ_(n)) and the axial stress (σz).

Adding and subtracting the Lubinski neutral stress (σ_(n)) in the first term on the right, defining Δσ_(z) as Lubinski's “excess axial stress above its neutral value (σ_(n)),” and re-arranging terms, the Triaxial Leak Criterion becomes:

${\sqrt{(\Delta)^{2} + \left( {\Delta\sigma_{z}} \right)^{2}} + {\frac{\alpha}{3}\left( {\Delta\sigma_{z}} \right)}} = {{{- \alpha}\sigma_{n}} + {\beta.}}$

Using the equation for the Lubinski neutral stress (σ_(n)), which is a function of the internal radius (a), external radius (b), internal pressure (Pi) and external pressure (Po), the Triaxial Leak Criterion becomes:

${\sqrt{(\Delta)^{2} + \left( {\Delta\sigma_{z}} \right)^{2}} + {\frac{\alpha}{3}\left( {\Delta\sigma_{z}} \right)}} = {{\alpha\frac{{b^{2}p_{o}} - {a^{2}p_{i}}}{b^{2} - a^{2}}} + {\beta{``{{Leak}{Load}{Equation}}"}}}$

where the internal radius (a) and external radius (b) refer to the dimensions of the threaded connection, not the pipe body. For alpha equals zero, the Leak Load Equation reduces to the equation for the von Mises circle. If the internal pressure (Pi) is equal to the external pressure (Po) and is equal to the pressure exerted by the drilling mud at a depth (mud density (ρ) times depth (z)), then the right side of the equation above becomes (α) (ρ) (z)+β. This demonstrates that leak resistance is dependent on depth (z). Finally, the Leak Load Equation also provides the basis for a graphical representation for leak assessment, as described herein.

Straight algebra can be applied here, as with other yield theories from the field of solid mechanics, to calculate the constants, alpha (α) and beta (β), from simple laboratory tests. The new Triaxial Leak Criterion is similar to the Drucker-Prager yield criterion, which is a three-dimensional, hydrostatic-pressure-dependent model for analyzing stresses, deformation, and failure in materials such as soils, concrete, and polymers. The Coulomb theory is a two-dimensional subset of the Drucker-Prager criterion and has been applied to model the stresses in sand, soils, concrete and similar materials.

The Coulomb theory includes two material constants: angle of internal friction (ϕ) and cohesion (c). When compared to the connection constants of a threaded connection, the angle of internal friction (ϕ) is analogous to alpha (α), which is associated with the geometry of the threads. The cohesion (c) is analogous to beta (β), which is associated with the components of the seal, including the effects of make-up interference and thread compound (for standard API connections) and metal-to-metal seals or elastomer rings with lubricants or other materials (for non-API proprietary connections). Applying the Coulomb failure criterion in the context of the Triaxial Leak Criterion provides a formula for the constants, alpha (α) and beta (β), in terms of the Coulomb constants:

$\alpha = \frac{2\sin\phi}{\sqrt{3}\left( {3 - {\sin\phi}} \right)}$ $\beta = {\frac{6c\cos\phi}{\sqrt{3}\left( {3 - {\sin\phi}} \right)}.}$

Alpha (α) is a function of the angle (ϕ) with no dependence on the cohesion (c). Beta (β) is a function of both the angle (ϕ) and the cohesion (c).

The determination of the connection constants, alpha (α) and beta (β), can be simplified by setting a number of boundary conditions which, in practice, are useful for designing experimental tests for measuring the values for alpha (α) and beta (β) for a particular connection. As described herein, simple laboratory leak tests can be performed to determine the values for the connection constants.

A FIRST EXAMPLE EXPERIMENT involves placing a threaded connection in a uniaxial testing state, which is characterized by an axial load only—with no internal or external pressure. The axial load includes tension or compression. Thus, the experiment involves two tests. First, exerting an axial force in tension on the threaded connection and measuring the axial tension leak stress (σt) at which leak occurs. Second, exerting an axial force in compression on the threaded connection and measuring the axial compression leak stress (σc) at which leak occurs. In each test, a small but insignificant pressure may be used, to measure or otherwise sense when leak occurs.

The differences in the uniaxial leak stresses generate asymmetry, as predicted by the Drucker-Prager model. The uniaxial asymmetry ratio (m) can be expressed as:

$m = {\frac{\sigma_{t}}{\sigma_{c}}.}$

In the context of leak testing, a leak will start when the uniaxial stress (σz) reaches the axial tension leak stress (σt) or the axial compression leak stress (σc). Because the radial stress and hoop stress are zero (since the internal and external pressures are zero), and applying the Leak Load Equation, the formulas for the connection constants can be expressed as: σ_(t)(1+α/3)=β σ_(c)(1−α/3)=β.

Solving for the connection constants:

$\alpha = {{3\frac{\sigma_{c} - \sigma_{t}}{\sigma_{c} + \sigma_{t}}} = {3\frac{1 - m}{1 + m}}}$ $\beta = {{2\frac{\sigma_{c}\sigma_{t}}{\sigma_{c} + \sigma_{t}}} = {{2\frac{\sigma_{t}}{1 + m}} = {2{\frac{m\sigma_{c}}{1 + m}.}}}}$

As shown, the constants, alpha (α) and beta (β), can be calculated based on an experiment that measures the axial tension leak stress (σt) and the axial compression leak stress (σc). This is fully analogous to the simple uniaxial tension test for determining yield strength (Y) for the pipe body, except two tests are required for the threaded connection because there are two constants.

The uniaxial asymmetry ratio (m) can be expressed as:

$m = {\frac{1 - \left( {\alpha/3} \right)}{1 + \left( {\alpha/3} \right)}.}$

Consideration of this formula for the uniaxial asymmetry ratio (m) demonstrates the role of alpha (α) and beta (β). If alpha is zero, then the ratio (m) equals one and the uniaxial leak limits for tension and compression are equal. (When alpha equals zero in the Triaxial Leak Criterion, the effect of the mean normal stress (P) is zero, and the von Mises stress equals beta). As alpha increases for a given beta, the axial tension leak stress (σt) decreases and the axial compression leak stress (σc) increases. For a given alpha, the two uniaxial leak stresses can only increase if beta increases.

These relationships confirm that the constant alpha (α) represents an internal property (namely, the geometry and behavior of the threads), and the constant beta (β) represents an external property (the seal elements, including makeup).

For a threaded connection, the constant alpha (α) is a dimensionless stress ratio that characterizes the thread behavior, as well as the taper, to transfer the load and displacement across the threads. In this aspect, the constant alpha (α) is a measure of the leak resistance ability of the threads and the taper. Accordingly, as used herein, the new connection constant alpha (α) is referred to as the Thread Modulus.

The constant beta (β) is related to the ability of the seal to resist leak. According to the equations above: when beta equals zero, both the axial tension leak stress (σt) and the axial compression leak stress (σc) equal zero. In other words, when there is no beta (i.e., no seal to resist leak), then no axial load can be supported by the connection without resulting in leak. In this aspect, the constant beta (β) is a measure of the leak resistance ability of the seal. Accordingly, as used herein, the new connection constant beta (β) is referred to as the Makeup Leak Resistance.

A SECOND EXAMPLE EXPERIMENT for testing a threaded connection involves placing a threaded connection in a pressurized testing state, which is characterized by known internal and external pressures—with zero axial load. As with the uniaxial tests, this experiment involves two tests (to determine the two connection constants). First, exerting an external test pressure (Po) on the threaded connection is used to measure the external leak pressure (Pb) at which external leak occurs. Second; exerting an internal test pressure (Pi) is used to measure the internal leak pressure (Pa) at which internal leak occurs.

For the external leak test, the axial load is zero, the internal pressure (Pi) is zero, and the external pressure (Po) is equal to the external leak pressure (Pb) at the outer radius (b). With these conditions, the Leak Load Equation after re-arranging terms becomes:

${P_{b}\left( {1 - \frac{\alpha}{3}} \right)} = {\beta\frac{b^{2} - a^{2}}{2b^{2}}}$ where the internal radius (a) and external radius (b) refer to the dimensions of the threaded connection.

For the internal leak test, the axial load is zero, the external pressure (Po) is zero, and the internal pressure (Pi) is equal to the internal leak pressure (Pa) at the inner radius (a). With these conditions, the Leak Load Equation after re-arranging terms becomes:

${P_{a}\left( {K + \frac{\alpha}{3}} \right)} = {\beta\frac{b^{2} - a^{2}}{2a^{2}}}$ where K is a geometric quantity based on the inner radius (a) and the outer radius (b), according to the equation:

${K = {\frac{1}{2}\sqrt{1 + {3\frac{b^{4}}{a^{4}}}}}}.$

Defining a pressure asymmetry ratio (n) is useful in solving for alpha (α) and beta (β) in the leak pressure relations above (for Pa and Pb). The pressure asymmetry ratio (n) is based on the internal leak pressure (Pa) at which leak occurs, and the external leak pressure (Pb) at which leak occurs, according to the equation:

${n = {K\frac{a^{2}P_{a}}{b^{2}P_{b}}}}.$

Using the pressure asymmetry ratio (n), alpha (α) and beta (β) can be determined. The Thread Modulus (α) is calculated according to the equation:

${\alpha = {3K\frac{1 - n}{K + n}}}.$

The Makeup Leak Resistance (β) is calculated according to the equation:

${\beta = {{2{P_{b}\left( \frac{n\left( {K + 1} \right)}{K + n} \right)}\left( \frac{b^{2}}{b^{2} - a^{2}} \right)} = {2{P_{a}\left( \frac{K\left( {K + 1} \right)}{K + n} \right)}\left( \frac{a^{2}}{b^{2} - a^{2}} \right)}}}.$

As described herein, the Thread Modulus (α) and the Makeup Leak Resistance (β) can be measured objectively by experimentation—involving only two tests. The constants (α, β) for a particular threaded connection do not change. Therefore, the constants (α, β) measured using the pressurized testing state will be identical to the constants (α, β) measured using the uniaxial testing state. Other test conditions may be applied to a threaded connection, resulting in the same measurements for the constants (α, β).

A threaded connection can withstand differential pressures higher than current API ratings because of the effects of hydrostatic pressure. Internal and external pressures close the connection and “energize” (enhance) the seal; axial loads open or “dilate” the connection and reduce the seal. Formulated from the new Triaxial Leak Criterion, a leak line and a leak circle are also introduced for assistance in tubular design, to help designers easily and quickly identify critical loads for leak and, together with the leak safety factor plot versus depth (see FIG. 3 ) for one or more critical load cases, determines the location where leak is most likely to occur along the string.

The leak line and leak circle are more comprehensive and more accurate than existing leak envelopes because the new Triaxial Leak Criterion applies to all combinations of internal pressure, external pressure, and axial tension or compression—as opposed to the simple load combinations (without backup pressure) derived from the currently used lab tests or simulations using finite-element analysis. Additionally, the new leak line and leak circle plots can be easily developed for both API connections and non-API proprietary connections.

LEAK LINE: The Triaxial Leak Criterion is written in the classical form of a linear equation; namely, y=mx+b, where m is the slope of the line and b is the y-axis intercept. Accordingly, on a graph where the horizontal or x-axis represents values for mean normal stress (P) and the vertical or y-axis represents values for von Mises stress (σ_(VM)), the slope of the leak line 200 is alpha (α, the Thread Modulus) and the y-intercept is beta (β, the Makeup Leak Resistance), as shown in FIG. 1 .

FIG. 1 is a graphical representation of the Triaxial Leak Criterion 25 for a particular threaded connection. The leak line 200 is plotted according to the Triaxial Leak Criterion 25, which is displayed in FIG. 1 for reference.

The Thread Modulus (α) and the Makeup Leak Resistance (β) can be determined through experimentation, as described herein. These constants (α, β) do not change. The leak line 200 for a threaded connection will always have a slope equal to the Thread Modulus (α) and a y-intercept equal to the Makeup Leak Resistance (β). In general, the leak risk is high for data points plotted on the graph that appear near or above the leak line 200; the leak risk is low for points significantly below the leak line 200.

In practice, the same threaded connection is used for all pipe joints in a well string. This minimizes the risk of error in the field during assembly. Accordingly, a tubular typically consists of the same OD, ID, grade (yield strength), and threaded connection along its entire length. Sometimes a tubular has two or more sections, wherein each section has the same parameters (OD, ID, grade (yield strength), and threaded connection) throughout the section.

The leak line 200 (and leak circle, described herein) apply to a section which has the same parameters. Accordingly, the Thread Modulus (α) and the Makeup Leak Resistance (β) are the same at all points in the section. Therefore, along the length of a section of the string, the only variables are the set of conditions—internal pressure (Pi), external pressure (Po), and axial force (Fz)—for a given load case.

A load case, as used herein, refers to a set of conditions (pressures and forces) that are expected to occur or may occur in the field during the lifetime of the well. Different load cases are based on different load types, such as evacuation or gas kick. A load case includes a profile of conditions—internal pressure (Pi), external pressure (Po), and axial force (Fz)—at each point (depth, z) along the string (or section of a string). Those load values can be used to calculate the mean normal stress (P) and the von Mises stress (σ_(VM)) at each depth (z) along the string, for each load case.

For example, in FIG. 1 , the locus of loads 401 on the graph is a plot of the pair of values (P, σ_(VM)) at every depth (z) along a string for a first load case. The locus of loads 402 shows the values for a second load case; and so on. A locus of loads may be plotted for several different load cases, as shown in FIG. 1 , which provides the designer with a visual tool for assessing the relative leak risk among different load cases.

Each locus of loads is plotted as a continuous line because a connection can be situated anywhere along the tubular string. For analytical purposes, the fact that the threaded connections are spaced apart is not important. The precise location of each threaded connection is typically not part of the well planning process because the actual location, in the field, can and will vary.

The curvilinear line for each locus of loads represents the relative leak risk of a particular threaded connection at each location along the string, relative to the leak line 200. If the locus of loads for any load case is near or above the leak line 200, then the leak risk for that load case is high for a string that uses that particular threaded connection. The designer may also refer to the plot of the leak safety factor versus depth, for that load case, to determine the location (or locations) along the string where the leak risk is high.

If the leak risk for any load case is higher than acceptable, the well designer may select a second threaded connection for use; and then repeat the steps of calculating and plotting the pairs of values (P, σ_(VM)) for various load cases. Note: the leak line will be different because the second threaded connection has its own unique Thread Modulus (α) and Makeup Leak Resistance (β) (and the outer diameter of the second threaded connection may also be different). In this aspect, the leak line graph shown in FIG. 1 provides a tool for designers to evaluate different threaded connections with different load cases.

LEAK CIRCLE. Like the leak line 200 shown in FIG. 1 , the leak circle 600 shown in FIG. 2 applies to a string of pipe with the same threaded connection. The graph in FIG. 2 is similar to the von Mises circle, except the formulas for each axis have been adjusted in accordance with the new Triaxial Leak Criterion 25. If the von Mises circle axes were used without adjustment, then a different circle radius for each depth and each load case would be required.

The fluid pressures (Pi, Po) acting on a threaded connection vary with depth (z). The Leak Load Equation is an expression of the Triaxial Leak Criterion which includes internal and external pressures (Pi, Po). By assuming that the pressures (on the right side of the Leak Load Equation) at each depth (z) along the string can be approximated for all load cases by an equivalent fluid density at depth (z), and by using the internal and external densities (pi, po), the Leak Load Equation becomes:

${\sqrt{(\Delta)^{2} + \left( {\Delta\sigma_{z}} \right)^{2}} + {\frac{\alpha}{3}\left( {\Delta\sigma_{z}} \right)}} = {{\alpha z\frac{{b^{2}\rho_{o}} - {a^{2}\rho_{i}}}{b^{2} - a^{2}}} + \beta}$ where Δ

is the differential pressure and Δσ_(z) is the excess axial stress. The right side of the above equation defines the leak resistance at death (z):

${L_{z} = {{\alpha z\frac{{b^{2}\rho_{o}} - {a^{2}\rho_{i}}}{b^{2} - a^{2}}} + \beta}}{\text{“Leak   Resistance   at   Depth”}\text{“z”}}$ which is displayed (16) on FIG. 2 for reference. Dividing the preceding equation by (Lz) on both sides provides the dimensionless result:

${\sqrt{\left( {\Delta/L_{z}} \right)^{2} + \left( {\Delta\sigma_{Z}/L_{z}} \right)^{2}} + {\frac{\alpha}{3}\left( {\Delta\sigma_{z}/L_{z}} \right)}} = {1.}$

Completing the square and re-arranging terms provides the formulas for the axes for the graph in FIG. 2 ; namely, Δ

* for the y-axis and Δσ_(z)* for the x-axis:

Δ * = ( 1 - ( α / 3 ) 2 ) ⁢ Δ𝒫 L z ⁢ Δσ z * = ( 1 - ( α / 3 ) 2 ) ⁢ Δ ⁢ z L z where Δ

* is the normalized differential pressure and Δσ_(z)* is the normalized excess axial stress. These normalized values are dimensionless because of (Lz). These values are described as ‘normalized’ because, compared to Δ

and Δσ_(z), the normalized values include the effect of the Thread Modulus (α) and the Makeup Leak Resistance (β).

Values for normalized differential pressure (Δ

*) are displayed on the y-axis of the graph in FIG. 2 . Values for normalized excess axial stress (Δσ_(z)*) are displayed on the x-axis. The normalized differential pressure equation 18 is displayed in FIG. 2 for reference. The normalized excess axial stress equation 19 is also displayed in FIG. 2 for reference. Then, (Δ

*)²(Δσ*_(z)+α/3)²=1 is the formula for the leak circle 600, which is centered on the x-axis at Δ

*=0 and Δσ_(z)*=−α/3, and has a radius equal to one, as shown in FIG. 2 .

If alpha equals zero, the leak circle reduces to the von Mises circle, which has a radius of one and is centered at the origin of the axes, which are ΔP* and Δσ_(z)* of the leak circle 600 as expressed above.

FIG. 2 is a graphical representation of the leak circle 600 at a particular depth (z) along the string. Each depth (z) along the string will have its own leak circle. Each load case involves the same threaded connection and has a pair of values (ΔP*, Δσ_(z)*), calculated using the equations above. Plotting the pair of values on the graph will display a load case point 701, which represents the relative leak risk at depth (z) for a particular load case (and a string with a particular threaded connection) relative to the leak circle 600.

The other load case points represent the pair of values (ΔP*, Δσ_(z)*) for other load cases. The plot in FIG. 2 shows six different load cases relative to the leak circle 600. This graph helps the designer select the best threaded connection to withstand all load cases without leaking.

The leak risk is low for load case points plotted significantly inside the leak circle 600; points near or outside the leak circle 600 have a high leak risk. The closer a point is to the leak circle 600, the higher the risk of leak (for that load case, and connection).

LEAK SAFETY PLOT: If one or more load case points are near the leak circle 600 in FIG. 2 (and/or one or more locus of loads 401 is near the leak line 200 in FIG. 1 ), the designer may select one or more load cases for additional scrutiny. FIG. 3 is a plot of the leak safety factor (SF_(Leak)) on the x-axis versus depth (z) on the y-axis for a particular string. The leak safety equation 30 is displayed for reference.

FIG. 3 includes a line representing a threshold value 500. Typically, the threshold value 500 represents a leak safety factor that is equal to or greater than one. An additional ‘design factor’ may be selected by the designer, as an additional margin of safety. For example, a designer or operator may specify a threshold value 500 equal to 1.25 (which is the number typically used for the pipe body).

One or more load cases may be selected for display on the leak safety plot shown in FIG. 3 . Each string will have its own leak safety plot. For each load case selected, the value of the leak safety factor (SFLeak) can be calculated for each depth (z) along the string and plotted, as shown in FIG. 3 . Five different load case lines are shown.

The closer the line is to the threshold value 500, the higher the leak risk. The load case lines provide a visual representation of the relative leak risk, which may include load cases that do not meet the threshold value 500 for leak safety. For example, the load case line 710 crosses the line representing the threshold value 500, indicating a high risk of leak. The leak safety plot also displays the depth at which leak is expected to occur for a load case. Load case line 710, for example, is expected to leak at or near a critical depth (zc).

The load case lines also provide a visual representation of the leak risk relative to other load cases. For example, load case line 702 indicates a higher safety factor compared to load case line 710 at all depths. In this aspect, the graph may be used for a visual assessment of a plurality of load cases, beginning with a first load case and first load case line, and then subsequently evaluating and plotting other load case lines.

Just as one characteristic constant (yield strength, Y) has been used historically to evaluate the mechanical limits of the pipe-body material, the two characteristic connection constants described herein—the Thread Modulus (α) and the Makeup Leak Resistance (β)— may be used to evaluate the mechanical limits for leak of threaded connections. Without the new Triaxial Leak Criterion described herein, well designers, operators, and connection suppliers will continue to use: (a) expensive testing methodology and FEA to qualify threaded connections, and (b) approximate safety factors, when selecting connections for specific applications. Neither (a) nor (b) currently consider hydrostatic dependence of connection performance. The new Triaxial Leak Criterion with two constants introduces the dependence on hydrostatic pressure, which is fundamental and different from all prior connector theories. The new Triaxial Leak Criterion has a key assertion—mechanical failure in all threaded connectors depends on hydrostatic pressure.

The technology disclosed herein may be used to reduce qualification testing and computer stress analysis (FEA) of threaded connections—with more efficient extrapolation across different connection types and sizes. The technology may also facilitate the optimized selection and placement of lower-cost threaded connections, resulting in structural efficiencies and cost savings. The technology may also facilitate the development and design of new types of threaded (and non-threaded) connections to optimize the connection constants, alpha (α) and beta (β). Further, the technology may facilitate the development and use of new leak safety factors for connections, which will lead to more reliable and cost-efficient well designs. This technology is applicable to not only well designs, but also to any application in any industry that requires efficient performance of conduits with connections that have separable male-female parts (not welded) and that seal between internal and external environments. Even for non-sealing applications, such as solid (or non-solid) rods or columns with threaded connections, the new Triaxial Leak Criterion with dependence on the mean normal stress applies to structural failure of the threaded connections, such as from pullout or thread fracture.

Connection Stresses at the Pin-Box Interface: The leak behavior of a threaded connection under triaxial stress is fully described by the triaxial connection leak criterion, as described herein. σ_(VM) =−αP+β  “Triaxial Leak Criterion 25”.

For the Triaxial Leak Criterion 25, the von Mises stress (σ_(VM)) is a linear function of the mean normal stress (P) in which the slope is alpha (α, the Thread Modulus) and the y-intercept is beta (β, the Makeup Leak Resistance). FIG. 1 is a graphical representation of an example leak line 200 plotted according to the Triaxial Leak Criterion 25 for an example threaded connection.

For a pipe body with no bending, the von Mises stress (σ_(VM)) is equal to the yield strength Y of the pipe material. In practice, the yield strength Y is determined from a uniaxial tensile test on a small rod specimen until it yields. For both types of yield (e.g., burst and collapse) the von Mises stress (σ_(VM)) is at its maximum at the inner diameter (ID). The leak must pass through the pipe body at the ID whether it originates internally or externally.

For threaded connections, the pin and box are offset axially, which means that an axial force F will generate a thread shear stress at the pin-box interface.

FIG. 4 is a cross-sectional view of an example tubing or casing coupled connection 400 (e.g., API or premium) having a radial interference seal. This view of an example connection 400 includes an upper pin 410, a lower pin 412, and a box 420. The conditions, as shown, include the axial force (F), internal pressure (Pi), and external pressure (Po), each of which is known or can be measured or calculated. A_(pin) represents the cross-sectional area of the pin 410, 412 which is exposed to the internal pressure (Pi). A_(box) represents the cross-sectional area of the box 420 which is exposed to the external pressure (Po).

The radius (a) is correlated with the inner diameter (ID) of the threaded connection, where the calculations were conducted for a pipe body with no bending. The radius (c) is correlated with the outer diameter of the connection. The radius (b) represents the radius associated with the pin-box interface. In some example implementations, the pin-box interface radius (b) is correlated with the pitch diameter (i.e., the diameter where the thread thickness is equal to the space between the threads). For some threads, the pitch diameter coincides with the geometric center of the thread flank (the sloping side of the thread). In some implementations, the pin-box interface radius (b) is equal to the radius on the pitch line at a specified location (distance from pin end) when the connection is hand-tight. The pitch diameter is a known value provided by the manufacturer or it can be reasonably estimated. For example, for API connections, the pitch diameter is specified at the location L1 for API 8-round or L7 for API buttress.

FIG. 5 is a cross-sectional view of an example drill string connection 450 having an axial interference shoulder seal. This view of an example connection 450 includes a pin 510 and a box 520. Ai represents the cross-sectional area of the pin 510 which is exposed to the internal pressure (Pi). Ao represents the cross-sectional area of the box 520 which is exposed to the external pressure (Po).

There are three stress mechanisms common to all tubular threaded connections, including the example connections 400, 450 shown in FIG. 4 and FIG. 5 , respectively.

First; the seal design impacts the stresses generated in a threaded connection. The Triaxial Leak Criterion 25 for connections, α_(VM) =−αP+β assumes that the seal and makeup behavior is completely characterized by the two constants, alpha and beta, which are different for different connections. The actual sealing mechanism is complex and varies, including, for example, thread seals, elastomer seals, or metal-to-metal seals (for connections like the example connection 400 in FIG. 4 ) or rotary shoulder seals (for connections like the example drill string connection 450 in FIG. 5 ).

Second; because of the seal, hydrostatic pressure is a key driver for evaluating leak resistance. Without a seal, the pressures (internal and external) would penetrate the threads and load the pin and box equally on all surfaces. When a seal is present, even an open-ended connector submerged in a pressure chamber would experience shear.

Finally; because the pin and box are offset axially, the axial force F will generate thread shear. Thread shear generates dilatancy associated with flank angles, meaning the connection opens, decreasing leak resistance. The dilatancy behavior is similar to that in sand and soils which dilate when sheared. Even without an axial force F, the internal pressure Pi alone generates a pin-end force, as shown in FIG. 4 and FIG. 5 .

For axisymmetric stresses with longitudinal thread shear stress (τ) (tau), the von Mises stress (σ_(VM)) is calculated from the expression:

$\sigma_{VM} = {\sqrt{{\frac{1}{2}\left\lbrack {\left( {\sigma_{r} - \sigma_{\theta}} \right)^{2} + \left( {\sigma_{\theta} - \sigma_{z}} \right)^{2} + \left( {\sigma_{z} - \sigma_{r}} \right)^{2}} \right\rbrack} + {3\tau^{2}}}.}$

For casing connections, like the example connection 400 in FIG. 4 , the radial stress (σr) and the hoop stress (σθ) at the pin-box interface radius (b) are determined using constants A and B from the Lame equations:

${\sigma_{r} = {A - \frac{B}{b^{2}}}}{\sigma_{\theta} = {A + \frac{B}{b^{2}}}}{{A = \frac{{a^{2}P_{i}} - {c^{2}P_{o}}}{c^{2} - a^{2}}},{B = \frac{a^{2}{c^{2}\left( {P_{i} - P_{o}} \right)}}{c^{2} - a^{2}}}}$ and it follows that,

$\frac{\sigma_{r} + \sigma_{\theta}}{2} = {A = \sigma_{n}}$ where A or σn is the neutral axial stress, as defined by Lubinski, and represents the mean normal stress in the connection cross-section.

Using the equations above for the von Mises stress (σ_(VM)) and the mean normal stress (σn), a new Leak Criterion with Thread Shear 425 in some implementations is expressed as:

${{\sqrt{\left( {\sqrt{3}\Delta{p\left( \frac{c^{2}}{c^{2} - a^{2}} \right)}\frac{a^{2}}{b^{2}}} \right)^{2} + \left( {\Delta\sigma_{z}} \right)^{2} + {3\tau^{2}}} + {\frac{\alpha}{3}\Delta\sigma_{z}}} = {{{- \alpha}\sigma_{n}} + \beta}}.$

In practice, as described herein, a designer can approve a threaded connection if the leak safety factor based on von Mises stress with thread shear is greater than a threshold value, which is typically a number equal to or greater than one. An additional ‘design factor’ may be selected by the designer (or imposed by an operator) as an additional margin of safety, in order to account for manufacturing tolerances and the like. For example, a designer or operator may specify 1.25 for the threshold value.

The Leak Criterion with Thread Shear 425 generates a circle when alpha is zero and the shear stress (tau) is zero. When plotted on a graph in which the abscissa value (along the x-axis) is the effective stress (Δσz=σz−σn) and the ordinate value (along the y-axis) is the differential pressure (Δp) quantity in the bracket, the radius of the circle is beta.

For a non-zero alpha and a shear stress (tau) of zero, the Leak Criterion with Thread Shear 425 generates a displaced circle with its center located at (negative-alpha divided by 3) on the dimensionless abscissa. The circle radius depends on the neutral axial stress (σn). Accordingly, the Leak Criterion with Thread Shear 425, applied over a series of threaded connections, would generate a series of displaced circles each having a different radius for each depth (z).

When the neutral axial stress (σn) is replaced with the equivalent expression in terms of differential pressure (Δp) and backup pressure, the Leak Criterion with Thread Shear 425 generates a customary skewed ellipse when plotted on axes of differential pressure (Δp) and axial stress (σz). For external leak with a backup pressure (Pi) the abscissa becomes (αz+Pi), which is like pipe collapse and demonstrates the dual and equal dependence on axial stress (σz) and backup pressure (Pi).

For the square-root quantity in the Leak Criterion with Thread Shear 425, the leak rating (Δp) decreases as the effective stress (Δσz=αz−αn) or the shear stress (tau) increases. This relationship means that leak is derated for axial force (F), backup pressure (Pi), and thread shear stress (tau). The API equation for internal leak does not include any of these effects.

The axial stress (σz) at any location in a threaded connection is not determined from axial force (F) alone. The internal pressure (Pi) exerts forces on the internal surfaces of the pin as shown in FIG. 4 and FIG. 5 . The external pressure (Po) exerts forces on the external surfaces.

The axial force (Fb) at the mid-point of the box may be expressed as: F _(b) =F+P _(i) A _(pin) −P _(o) A _(box) where A_(pin) represents the exposed end area of the pin and A_(box) represents the exposed end area of the box.

The average axial stress (σz) in the thread region may be expressed as:

$\sigma_{z} = {\frac{F - {P_{o}A_{box}}}{A_{con}} = \frac{F_{b} - {P_{i}A_{pin}}}{A_{con}}}$ where A_(con) represents the cross-sectional area of the connection. The connection area may be calculated using A_(con)=π(c²−a²).

The axial force along the surface of the pin varies from tension (F) at the pipe end (e.g., the lower end of the lower pin 412, as shown in FIG. 4 ) to compression (— Pi A_(pin)) at the nose end of the pin (e.g., the internal surface of the lower pin 412). This change in force along the pin equals [F−(−Pi A_(pin))], so the thread shear stress (tau) along the engaged thread length (L_(t)) may be expressed as:

$\tau = {\frac{F + {P_{i}A_{pin}}}{A_{t}} = \frac{F_{b} + {P_{o}A_{box}}}{A_{t}}}$ where A_(t) is the engaged thread area over which the shear stress (tau) acts. The engaged thread area may be calculated using A_(t)=2πbL_(t), where the engaged thread length (L_(t)) is a known value provided by the manufacturer or it can be measured or estimated. As expressed in the equation above, the shear stress (tau) acting on the pin is equal to the shear stress acting on the box. The force along the surface of the box varies from tension (i.e., a box force F_(b) acting at the box midpoint) to compression (−Po A_(box)) at the box end. The axial forces acting on the pin and box generate the shear stress (tau) between the pin and box.

For known values of force (F), internal pressure (Pi), and external pressure (Po), the three normal stresses and the thread shear stress (tau) may be calculated, starting with the Lame equations above.

The von Mises stress (σ_(VM)), as described above, may be calculated from the expression:

$\sigma_{VM} = {\sqrt{{\frac{1}{2}\left\lbrack {\left( {\sigma_{r} - \sigma_{\theta}} \right)^{2} + \left( {\sigma_{\theta} - \sigma_{z}} \right)^{2} + \left( {\sigma_{z} - \sigma_{r}} \right)^{2}} \right\rbrack} + {3\tau^{2}}}.}$

The mean normal stress (P), as described above, may be calculated from the expression:

$P = {\frac{\sigma_{r} + \sigma_{\theta} + \sigma_{z}}{3}.}$

Applying the Triaxial Leak Criterion 25, σ_(VM) =−αP+β the values of the two constants, alpha and beta, are needed in order to calculate the connection leak safety factor (SF_(C)):

${SF}_{c} = \frac{{{- \alpha}P} + \beta}{\sigma_{VM}}$ in which the numerator represents the working limit, and the denominator represents the working stress. In practice, as described herein, a designer can approve a threaded connection if the leak safety factor is greater than a threshold value, which is typically a number equal to or greater than one. An additional ‘design factor’ may be selected by the designer (or imposed by an operator) as an additional margin of safety, in order to account for manufacturing tolerances and the like. For example, a designer or operator may specify 1.25 for the threshold value.

The two constants, alpha and beta, can be determined using one or more tests in which a number of values and conditions are known (or readily measured).

First Test Method: For example, two leak tests conducted on a known connection exposed to uniaxial force (F), no internal pressure (Pi=0), and no external pressure (Po=0) can be used to determine the constants, alpha and beta.

Using an API 7-inch 35-ppf N80 LTC connection as an example case, the following data are known: inner radius (a)=3.002 inches; pin-box interface radius (b)=3.452 in.; outer radius (c)=3.9375 in.; engaged thread length (L_(t))=3.296 in.; exposed pin area (A_(pin))=7.164 in²; and exposed box area (A_(box))=11.023 in². Finite-element analysis (FEA) predicts that the N80 connection will open sufficiently to leak when the force in tension (T)=620 kips (i.e., 620,000 lbf) or when the compression (C)=690 kips. Note: the difference between the tensile and compressive limits demonstrates the hydrostatic effect (even when there is no applied internal or external pressure). The mean normal stress (P), using the expression above, when only axial stress (σz) is present, may be calculated using the expression, P=σz/3, which further demonstrates that axial stress alone is sufficient to cause the hydrostatic effect.

The difference between the uniaxial tensile and compressive limits also demonstrates that the von Mises stress equation alone, with only a single constant, would be insufficient to characterize connection leak risk because the von Mises equation requires the uniaxial limits to be equal in magnitude. FEA analysis confirms the difference in limits. The Triaxial Leak Criterion 25 demonstrates that two constants are required to characterize a connection leak risk.

A uniaxial asymmetry ratio (m) can be used to determine the constants, alpha and beta.

$m = {\frac{\sigma_{T}}{\sigma_{C}} = \frac{T}{C}}$ where the tensile stress (σ_(T)) equals the tension (T) divided by the cross-sectional area (A_(con)) of the LTC connection, and the compressive stress (σ_(C)) equals the compression (C) divided by the cross-sectional area (A_(con)) of the LTC connection.

In laboratory testing, a uniaxial leak test can be conducted with either a small amount of pressure or with some non-mechanical sensor. For uniaxial test conditions with a small pressure for leak detection, internal pressure (Pi) is considered zero and external pressure (Po) is considered zero; therefore, the differential pressure (Δp) is zero and the neutral axial stress (σn) is zero.

For uniaxial tension, the tensile leak limit is reached when the uniaxial stress (σz) reaches the tensile stress (σ_(T)). For tension, the Leak Criterion with Thread Shear 425 reduces to:

${\sigma_{T}\left\lbrack {\sqrt{1 + {3\left( \frac{A_{con}}{A_{t}} \right)^{2}}} + \frac{\alpha}{3}} \right\rbrack} = {\beta.}$

For uniaxial compression, the compressive leak limit is reached when the uniaxial stress (σz) reaches the compressive stress (negative (−σ_(C))). For compression, the Leak Criterion with Thread Shear 425 reduces to:

${\sigma_{C}\left\lbrack {\sqrt{1 + {3\left( \frac{A_{con}}{A_{t}} \right)^{2}}} - \frac{\alpha}{3}} \right\rbrack} = {\beta.}$

Defining a uniaxial test constant (S),

$S = \sqrt{1 + {3\left( \frac{A_{con}}{A_{t}} \right)^{2}}}$ and solving for alpha and beta, the Thread Modulus alpha (α) equals:

$\alpha = {3\frac{1 - m}{1 + m}S}$ and the Makeup Leak Resistance beta (β) equals:

$\beta = {{{\sigma_{T}\left( \frac{2}{1 + m} \right)}S} = {\left( \frac{T}{A_{con}} \right)\left( \frac{2}{1 + m} \right)S}}$ or $\beta = {{{\sigma_{C}\left( \frac{2m}{1 + m} \right)}S} = {\left( \frac{C}{A_{con}} \right)\left( \frac{2m}{1 + m} \right){S.}}}$

Solving for the uniaxial asymmetry ratio (m) from the equation above for alpha,

$m = \frac{S - \left( {\alpha/3} \right)}{S + \left( {\alpha/3} \right)}$ which means the ratio (m) is between zero and one (0≤m≤1) and the Thread Modulus alpha (α) is between zero and {three times S} (0≤α≤3S) in cases where alpha is positive. The value of alpha may be negative for certain load conditions and certain types of connections.

For the example LTC connection, and the FEA limits for tension (T=620 kips) and compression (C=690 kips), the ratio (m) is 0.899 and the uniaxial test constant (S) is 1.115. According to the equations above, the Thread Modulus alpha (α) is 0.179 and the Makeup Leak Resistance beta (β) is 35,720 psi.

Although the values for alpha and beta were calculated based on the example uniaxial test conditions described above, the Thread Modulus alpha (α) and the Makeup Leak Resistance beta (β) are considered to be constant values associated with the example LTC connection, under any conditions, including field conditions. Accordingly, the constant values for alpha and beta are expected to be the same under different test conditions.

Second Test Method: In another example test, the two constants, alpha and beta, can be determined when a number of values and conditions are known or readily measured. In this second test method, two pressure tests are used. An internal leak test is conducted on a known connection exposed to a zero uniaxial force (F=0), no external pressure (Po=0), and a known internal leak pressure (Pi=Pa). The internal pressure (Pi) equals the leak pressure (Pa) at the inner radius because internal leak would occur at the inner radius (a). Under these conditions, the axial stress (σz) in the thread region is zero.

An external leak test is conducted on a known connection exposed to a zero uniaxial force (F=0), no internal pressure (Pi=0), and a known external leak pressure (Po=Pc). The external pressure (Po) equals the external leak pressure (Pc) at the outer radius because external leak would occur at the outer radius (c). Under these conditions, the shear stress (tau) is zero because the force is zero (F=0) and the internal pressure is zero (Pi=0).

The equations for internal and external leak can be simplified by using and calculating a number of pressure-test constants (K₀, K₁, K₂, and K₃) which are unitless and based on the geometry of the connection:

$K_{0} = {1 - {\left( \frac{c^{2} - a^{2}}{c^{2}} \right)\left( \frac{A_{box}}{A_{con}} \right)}}$ $K_{1} = {\frac{1}{2}\sqrt{{3\left( \frac{a}{b} \right)^{4}} + \left( K_{0}^{2} \right)^{4}}}$ $K_{2} = {\frac{1}{2}\left( {3 - K_{0}} \right)}$ $K_{3} = {\frac{1}{2}{\sqrt{1 + {3\left( \frac{c}{b} \right)^{4}} + {3\left( \frac{A_{pin}}{A_{t}} \right)^{2}\left( \frac{c^{2} - a^{2}}{a^{2}} \right)^{2}}}.}}$

Recall, the Leak Criterion with Thread Shear 425 described above:

${\sqrt{\left( {\sqrt{3}\Delta{p\left( \frac{c^{2}}{c^{2} - a^{2}} \right)}\left( \frac{a^{2}}{b^{2}} \right)} \right)^{2}} + \left( {\Delta\sigma}_{z} \right)^{2} + {3\tau^{2}} + {\frac{\alpha}{3}\Delta\sigma_{z}}} = {{{- \alpha}\sigma_{n}} + {\beta.}}$

Using the pressure-test constants (K₀, K₁, K₂, and K₃), the Leak Criterion with Thread Shear 425 for internal leak at the inner radius (a) reduces to:

${P_{a}{\frac{2a^{2}}{c^{2} - a^{2}}\left\lbrack {K_{3} + \frac{\alpha}{3}} \right\rbrack}} = \beta$ and the Leak Criterion with Thread Shear 425 for external leak at the outer radius (c) reduces to:

${P_{c}{\frac{2c^{2}}{c^{2} - a^{2}}\left\lbrack {K_{1} - {K_{2}\frac{\alpha}{3}}} \right\rbrack}} = {\beta.}$

Whereas a uniaxial asymmetry ratio (m) was applied above for the uniaxial force test, a unilateral asymmetry ratio (n) may be applied here for the pressure-test conditions, where n=(a² Pa)/(c² Pc).

Using the unilateral asymmetry ratio (n), the above equations can be solved for alpha and beta:

$\alpha = {3\frac{K_{1} - {K_{3}n}}{K_{2} + n}}$ $\beta = {{2P_{a}\frac{a^{2}}{c^{2} - a^{2}}\left( \frac{K_{1} + {K_{2}K_{3}}}{K_{2} + n} \right){or}{}\beta} = {2P_{c}\frac{c^{2}n}{c^{2} - a^{2}}{\left( \frac{K_{1} + {K_{2}K_{3}}}{K_{2} + n} \right).}}}$

For the example LTC connection, the equation for Leak Criterion with Thread Shear 425 and the equations for alpha and beta (above) predict that internal leak will occur when the pressure (Pa) reaches 9,975 psi and that external leak will occur when the external leak pressure (Pc) reaches 10,770 psi. Using these predicted internal leak pressure (Pa) and the external leak pressure (Pc), along with the geometry data for the example LTC connection in the above equations, confirms that the Thread Modulus alpha (α) equals 0.179 and the Makeup Leak Resistance beta (β) equals 35,720 psi.

Finite-element analysis (FEA) predicts that the example LTC connection will develop internal leak when the pressure (Pa) reaches 9,942 psi, which is close to the predicted value of 9,975 psi. The API internal leak value for the example LTC connection is 11,790 psi.

FIG. 6 is a graphical representation of the predicted leak conditions for an example LTC connection. When plotted on a graph in which the abscissa value (along the x-axis) is the axial force (F) and the ordinate value (along the y-axis) is the differential pressure (Δp=Pi−Po), the predicted leak values form an ellipse.

As shown on the graph in FIG. 6 , the solid X represents the API internal leak value of 11,790 psi for the example N80 connection when the axial force is zero (F=0).

The values for internal leak are plotted above the x-axis, where the differential pressure (Δp) is positive. The values for external leak are plotted below the x-axis, where the differential pressure (Δp) is negative. The transitions between the upper and lower halves of the ellipses are not smooth because the mean normal stress (σ_(n)) is different for internal leak conditions versus external leak conditions. The triangles in FIG. 6 represent the values predicted by finite-element analysis (FEA).

The three ellipses plotted in FIG. 6 are based on the Leak Criterion with Thread Shear 425, using the alpha and beta values that were calculated using the formulas shown above.

The first ellipse is plotted using a solid line for internal leak (above the x-axis) and a dotted line for external leak (below the x-axis). As shown in the legend, the values for internal leak were calculated using an external pressure of zero (Po=0). The values for external leak were calculated using an internal pressure of zero (Pi=0). The solid triangles represent the values predicted by finite-element analysis (FEA) when the external pressure is zero (Po=0). As described above, FEA predicted that the LTC connection will open sufficiently to leak when the force in tension (T)=620 kips (e.g., a positive force (F) along the x-axis) or when the compression (C)=690 kips (e.g., a negative force). As shown, the first ellipse (solid line) crosses the positive y-axis very near the solid triangle. The first ellipse (dotted line) crosses the negative y-axis at minus 10,770 psi.

The second ellipse is plotted using a dashed line with a single dot for internal leak (above the x-axis) and a dashed line for external leak (below the x-axis). As shown in the legend, the values for internal leak were calculated using an external pressure of 5,000 psi. The values for external leak were calculated using an internal pressure of 5,000 psi. The open triangles represent the values predicted by finite-element analysis (FEA) when the external pressure is 5,000 psi. As shown, the open triangle for the compression limit (along the negative x-axis) is close to the value predicted (where the second ellipse crosses the x-axis). The open triangle for the tension limit (along the positive x-axis) is somewhat greater than the value predicted. The open triangle along the positive y-axis is close to the value predicted. The second ellipse (dashed line) crosses the negative y-axis at minus 10,630 psi.

The third ellipse is plotted using a dashed line with two dots for internal leak (above the x-axis) and a double-dotted line for external leak (below the x-axis). As shown in the legend, the values for internal leak were calculated using an external pressure of 10,000 psi. The values for external leak were calculated using an internal pressure of 10,000 psi. The shaded triangle represents the value predicted by finite-element analysis (FEA) when the external pressure is 10,000 psi. As shown, the shaded triangle along the positive y-axis is close to the value predicted. The third ellipse (double-dotted line) crosses the negative y-axis at minus 10,450 psi.

The three ellipses plotted in FIG. 6 are off-center, as expected, because of hydrostatic pressure. As backup pressure increases, the ellipse shifts to the left and the difference between the tension and compression limits increases (i.e., the curves are further apart) because of the (αP) term in the Triaxial Leak Criterion 25: σ_(VM) =−αP+β.

FIG. 7 is a graphical representation of the predicted failure conditions for the pipe-body alone (steel yield strength (Y) for the outer ellipse, shown using dashed lines) and for the connection alone (leak for the inner ellipse, shown using solid and dotted lines) for the example LTC connection. The abscissa values (along the x-axis) represent the axial force (F) and the ordinate values (along the y-axis) represent the differential pressure (Δp=Pi−Po).

For the pipe-body (the outer ellipse), the Von Mises equation is used with the yield strength (Y) of the N80 pipe material (Y=80,000 psi). For Δp=0, the pipe ellipse shows that both the compressive and tensile limits are the same, namely 814 kips (based on Y times the cross-sectional area of the example pipe body). Loads inside the pipe ellipse indicate that the pipe is safe.

For the connection (the inner ellipse), the Leak Criterion with Thread Shear 425 is used to calculate the values on the ellipse. The inner ellipse for the connection is mostly inside the outer pipe-body ellipse, which indicates the connection will leak before the pipe body fails.

For evaluating leak, predicted values inside the inner ellipse indicate a safe connection, whereas values outside the leak ellipse indicate the connection will fail.

The inner ellipse for the connection is mostly inside the outer pipe-body ellipse, except for a small section in quadrant three. The two ellipses cross near the API collapse limit (shown by the X), suggesting that the pipe-body will collapse before external leak occurs.

Although the various embodiments are described with reference to petroleum engineering and down-the-hole drilling and production for oil and gas, the methods and systems described herein may be applied in a variety of contexts. For example, the technology and solutions described herein may be readily applied to any use or application that includes a column (vertical, horizontal, or otherwise) of elements having one or more threaded connections, especially when such a column is positioned within any substance that may be characterized as a fluid or causing fluid-like pressures on the column, such as from cement, soil, and rock or generally from any solid that squeezes or confines the column.

FIG. 8 is a partial cross-sectional view of an example generic, premium threaded connection 650 which has been slightly modified to leak in tension before the pipe body fails. The connection 650 selected for analysis herein is a generic premium connection; a 9⅝-inch 40-ppf N80 coupled casing connection with an internal metal-to-metal seal, buttress-type threads, and a torque shoulder. The connection 650 was modified for use with finite element analysis (FEA) to determine the leak constants (e.g., the thread modulus (α) and the makeup leak resistance (β)) as described herein. For example, the connection 650 was modified to have a reduced seal contact at make-up so the connection 650 would leak under moderate to high loads before pipe yield (i.e., the pipe body von Mises stress is within the yield limit). Otherwise, the pipe body would yield before the connection leaks.

For conducting a finite element analysis (FEA) of the example premium connection 650, the N80 casing material was modeled using a bilinear stress-strain relationship with an ultimate tensile strength of 100 ksi at 10% strain. An elastic modulus of 30,000 ksi and a Poisson's ratio of 0.3 were used. Contact interaction was modeled between the pin and the box. Friction was not considered at the contact interface. An axial symmetry boundary condition was applied at the coupling center, where the boundary was fixed along the longitudinal direction but was allowed to move radially. Pressure access was assumed; internal pressure was applied on the pin-shoulder interface to the seal, and external pressure was applied in the thread region to the seal.

A finite element analysis (FEA) of the example connection 650 evaluated eleven (11) load cases that would result in leaks. The leaking test cases 700A are tabulated in FIG. 9 including the combination of loads (e.g., pressures and forces) which, based on the FEA, would be expected to result in leak of the example connection 650. The FEA predicted the tensile forces (F) which, given the pressures, would produce leaks at the connection 650. For example, for leaking test case 1 in FIG. 9 , the FEA predicted that a tensile force (F) equal to 621 kips (tension is positive and compression is negative) would produce a leak. The test cases 700A can be actual, physical tests conducted in a laboratory which leak or FEA tests that predict leak. In either case, the tests are designed such that the connection leaks before the pipe-end of the test specimen fails.

FIG. 9 also includes a number of calculated values associated with each leaking test case 700A. For example, each differential pressure value (Δp) (=Pi−Po) 910 is calculated by subtracting the external pressure (Po) from the internal pressure (Pi). The normal stresses (radial, hoop, axial), the shear stress (tau), the Lubinski neutral stress (σ_(n)), and each von Mises stress value (σ_(VM)) 930 were calculated using the equations discussed herein. The mean normal stress (P) is calculated as the average of the three normal stresses, as described herein.

FIG. 9 also includes an effective pressure value (Pe) 920 associated with each leaking test case 700A. The value for each effective pressure (Pe) (=P−σn) is calculated by subtracting the neutral stress (σn) from the mean normal stress (P). For example, leaking test case 5 has a mean normal stress (P) equal to −2.523 and a neutral stress (σn) of −10. The effective pressure (Pe) for leaking test case 5 equals P (−2.523) minus an (−10), which equals +7.477.

FIG. 10 is a graph 900 of the leaking test cases 700A tabulated in FIG. 9 . As shown, the graph 900 in FIG. 10 is a Cartesian coordinate system in which each abscissa value along the x-axis is an effective pressure 920 (Pe) (=P−σn) and each ordinate value along the y-axis is a von Mises stress 930. As shown, the plotted data values associated with each particular differential pressure (e.g., Δp=2, Δp=1, Δp=0, Δp=1, Δp=2) are grouped together because they are nearly overlapping. For example, starting at the left side of the plotted values, a differential pressure of Δp=2 is associated with one case; specifically, leaking test case 8 in FIG. 9 , in which the effective pressure Pe is 4.897 and the von Mises stress is 27.742.

The differential pressure of Δp=1 is associated with two cases; specifically, leaking test cases 6 and 7 in FIG. 9 . Note, the plotted values associated with the two cases of Δp=1 are nearly overlapping.

The differential pressure of Δp=0 is associated with five cases; specifically, leaking test cases 1-5 in FIG. 9 . Again, the plotted values associated with these five leaking test cases are nearly overlapping.

As shown in FIG. 10 , the grouping of plotted values associated with the eleven leaking test cases 700A has a quadratic dependence on effective pressure (Pe) (=P−σn) with three constants. Curve fitting refers to the process of constructing a curve or a mathematical function that best fits a series of data points. In most types of curve fitting, the data are fitted using a number of successive approximations. Linear or straight-line fitting is a common form in which the mathematical model is a linear expression (e.g., y=mx+b). When the data points indicate a quadratic dependence, such as those shown in FIG. 10 , the mathematical model is a quadratic expression (e.g., y=ax²+bx+c) that most closely fits the data.

The leak behavior of a threaded connection as a function of effective pressure (Pe) (=P−αn) is described by a new quadratic leak criterion 325: σ_(VM) =δP _(e) ² −αP _(e)+β

where the first constant is called the leak path factor (δ) and the other two constants are the thread modulus (α) and the makeup leak resistance (β), as described herein. The first constant is called the leak path factor (δ) because it appears to depend on the leak flow path and the differential pressure (Δp) or, more specifically, on internal leak versus external leak.

The three leak constants will be different for different connections. Because thread modulus (α) and leak path factor (δ) are multipliers of the load terms, these two constants in some cases may depend on connection type instead of size; so those constants may be the same for a family of connections having the same type. The makeup leak resistance (β) will be different for different sizes, but it may correlate with one or more geometric parameters for the same family of connections.

For the example leaking test cases 700A tabulated in FIG. 9 and shown on the graph 900 in FIG. 10 , the fitting quadratic expression 330 a has a leak path factor (δ) equal to 1.61 psi⁻¹, a unitless thread modulus (α) of 21.43 and a makeup leak resistance (β) of 94.37 psi. The fitting quadratic expression 330 a, as shown, produces a parabola 331 that approximately fits the plotted data points.

Any of a variety of curve fitting models and tools may be used to perform the process of fitting a quadratic expression to the data points. For example, applications with graphical tools, such as Microsoft Excel, may include one or more fitting models (e.g., exponential, linear, logarithmic, polynomial, and the like) for analyzing sets of data points. In some implementations, the method of fitting includes plotting the data points on a graph, as shown in FIG. 10 .

The quadratic leak criterion 325 can also be expressed, of course, as a leak safety factor 330, according to the equation:

${SF}_{Leak} = \frac{{\delta P_{e}^{2}} - {\alpha P_{e}} + \beta}{\sigma_{VM}}$

in which the numerator represents the working stress limit and the denominator represents the working stress. In practice, as described herein, a designer can approve a threaded connection (for the anticipated loads) if the leak safety factor is greater than a threshold value, which is typically a number equal to or greater than one. An additional ‘design factor’ may be selected by the designer (or imposed by an operator) as an additional margin of safety, in order to account for manufacturing tolerances and the like. For example, a designer or operator may specify a value of 1.25 or greater for the threshold value.

FIG. 11 is a graphical representation of a quadratic leak safety factor 330 for an example string of threaded connections, including example load case lines for a number of example load cases 700 b. The quadratic leak safety factor for each string is plotted on the x-axis, versus depth (z) on the y-axis, for each example string. Each string will have its own leak safety plot. The graph includes a line representing the threshold value 500 b. For each of the example load cases 700 b, the value of the leak safety factor can be calculated for each depth (z) along the string and plotted.

Five different load case lines are shown. The closer the line is to the threshold value 500 b, the higher the leak risk. The load case lines provide a visual representation of the relative leak risk, which may include load cases that do not meet the threshold value 500 b for leak safety. For example, the load case line 710 b crosses the line representing the threshold value 500 b, indicating a high risk of leak. The leak safety plot also displays the depth at which leak is expected to occur for a load case. Load case line 710 b, for example, is expected to leak at or near a critical depth (zc).

The load case lines also provide a visual representation of the leak risk relative to other load cases. For example, load case line 702 b indicates a higher safety factor compared to load case line 710 b (at all depths). In this aspect, this kind of graph may be used for a visual assessment of a plurality of load cases 700 b.

As used herein, the term “optimize” refers to and includes selecting a threaded connection that will withstand the assumed loads in the well planning or design phase. In addition, the term “optimize” also refers to and includes the management or adjustment of the loads (e.g., pressures and forces), during the operation phase, to ensure safety and to maintain connection integrity. For example, a threaded connection may have been selected, delivered, or installed. The methods and systems described herein may be applied to optimize the operation of the connection by changing or adjusting the loads that will be acting on a connection.

Except as stated immediately above, nothing that has been stated or illustrated is intended or should be interpreted to cause a dedication of any component, step, feature, object, benefit, advantage, or equivalent to the public, regardless of whether it is or is not recited in the claims.

It will be understood that the terms and expressions used herein have the ordinary meaning as is accorded to such terms and expressions with respect to their corresponding respective areas of inquiry and study except where specific meanings have otherwise been set forth herein. Relational terms such as first and second and the like may be used solely to distinguish one entity or action from another without necessarily requiring or implying any actual such relationship or order between such entities or actions.

Unless otherwise stated, any and all measurements, values, ratings, positions, magnitudes, sizes, and other specifications that are set forth in this specification, including in the claims that follow, are approximate, not exact. Such amounts are intended to have a reasonable range that is consistent with the functions to which they relate and with what is customary in the art to which they pertain. For example, unless expressly stated otherwise, a parameter value or the like may vary by as much as plus or minus ten percent from the stated amount or range.

In addition, in the foregoing Detailed Description, it can be seen that various features are grouped together in various examples for the purpose of streamlining the disclosure. This method of disclosure is not to be interpreted as reflecting an intention that the claimed examples require more features than are expressly recited in each claim. Rather, as the following claims reflect, the subject matter to be protected lies in less than all features of any single disclosed example. Thus, the following claims are hereby incorporated into the Detailed Description, with each claim standing on its own as a separately claimed subject matter.

While the foregoing has described what are considered to be the best mode and other examples, it is understood that various modifications may be made therein and that the subject matter disclosed herein may be implemented in various forms and examples, and that they may be applied in numerous applications, only some of which have been described herein. It is intended by the following claims to claim any and all modifications and variations that fall within the true scope of the present concepts. 

What is claimed is:
 1. A method of evaluating leak safety, comprising: calculating a leak safety factor (SF_(Leak)) associated with a threaded connection as a function of an effective pressure (Pe) and a von Mises stress (σ_(VM)), according to the equation: ${SF}_{Leak} = \frac{{\delta P_{e}^{2}} - {\alpha P_{e}} + \beta}{\sigma_{VM}}$ wherein a leak path factor (δ), a thread modulus (α), and a makeup leak resistance (β) are constants associated with the threaded connection; and using the threaded connection when the calculated leak safety factor is greater than a threshold value.
 2. The method of claim 1, wherein the effective pressure (Pe) is defined by a mean normal stress (P) minus a neutral axial stress (σn).
 3. The method of claim 1, wherein the threshold value is a number equal to or greater than one.
 4. The method of claim 1, further comprising: calculating a leak risk associated with the threaded connection according to a quadratic leak criterion: σ_(VM) =δP _(e) ² −αP _(e)+β.
 5. The method of claim 1, wherein calculating the leak safety factor further comprises: generating at least three leaking test cases, each having a different differential pressure value Dp (=Pi−Po), and each characterized by a combination of loads expected to result in leak at the threaded connection; for each of the leaking test cases, calculating a subset of values comprising an effective pressure value (Pe) and a von Mises stress value (σ_(VM)); and fitting a quadratic expression to the subset of values, wherein the fitted quadratic expression comprises values for the leak path factor (δ), the thread modulus (α), and the makeup leak resistance (β).
 6. The method of claim 5, further comprising: plotting on a graph the at least three leaking test cases, each represented by its respective effective pressure value (Pe) and its respective von Mises stress value (σ_(VM)); and fitting the quadratic expression to the plotted leaking test cases, wherein the quadratic expression when plotted on the graph produces a parabola. 